a relation on

USAGE SUMMARY

The phrase "a relation on" is correct and usable in written English.
It is typically used in mathematical or formal contexts to describe a relationship defined on a set or between sets. Example: "In set theory, a relation on a set A is a subset of the Cartesian product A x A."

✓ Grammatically correct

Science

Human-verified examples from authoritative sources

Exact Expressions

18 human-written examples

A FOREACH operator with a "flatten" clause can expand one input tuple into a tuple list representing a relation on which relational operators can operate.

Data Science and Engineering

Pig operators process the whole data set that can be treated as a relation on which relational operators are applied naturally.

Data Science and Engineering

Thus, even if self-identity or God's relation to the world is a mere being of reason, and hence does not actually relate its subject to anything, nonetheless it can still be regarded as a relation on the grounds that it is signified by a relational concept.

SEP

For simplicity, assume that causation is a relation on events.

SEP

Let, and consider a relation on as follows: (2.35).

Fixed Point Theory and Applications

Define a relation on as follows: if and only if either or holds.

Fixed Point Theory and Applications

Human-verified similar examples from authoritative sources

Similar Expressions

42 human-written examples

Let be a binary relation on a set X and let A be a nonempty subset of X.

Fixed Point Theory and Applications

In particular, a ternary relation on a set is an AR-relation if and only if it satisfies axioms R1 R6; i.e., it is a disjunctive R-relation.

Bulletin of Mathematical Sciences

A ternary relation on a set is an R-relation if and only if it satisfies conditions R1 R5; i.e., it is basic and strongly transitive.

Bulletin of Mathematical Sciences

A binary relation on a set $A$ is a set of ordered pairs of elements of $A$, that is, a subset of $A\times A$.

SEP

So, if Predecessor is a one-to-one relation, it is a one-to-one relation on the natural numbers.

SEP

Expert writing Tips

Best practice

When using "a relation on", ensure the context clearly defines the set or domain upon which the relation is established. This is particularly important in mathematical or formal settings to avoid ambiguity.

Common error

Avoid using "a relation on" when you actually mean a relation between two distinct sets. "On" implies a relation within a single set, whereas "between" indicates a connection between two or more separate entities.

Linguistic Context

The phrase "a relation on" functions as a prepositional phrase typically used to specify the domain or set on which a mathematical or logical relation is defined. It indicates the scope of the relation. As Ludwig AI confirms, the phrase is grammatically correct.

Expression frequency: Uncommon

Frequent in

Science

88%

Encyclopedias

6%

News & Media

6%

Less common in

Formal & Business

0%

Wiki

0%

Reference

0%

Ludwig's WRAP-UP

In summary, "a relation on" is a grammatically correct prepositional phrase predominantly used in formal, scientific, and mathematical contexts. Its primary function is to define the scope or domain within which a certain relationship exists. While it may not be as common in everyday language, its precision makes it indispensable in specific fields. Ludwig AI confirms that this phrase is well-accepted and provides helpful examples for appropriate usage. If a more general expression is desired, alternatives like "a relationship concerning" or "a connection applicable to" can be considered to avoid using "a relation on".

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

a relationship concerning

This phrase uses "concerning" to indicate the subject of the relationship, which slightly alters the focus compared to "on".

a dependency with regard to

This alternative highlights the aspect of dependency, adding a level of specificity that "a relation on" may not always convey.

a connection applicable to

This alternative emphasizes the applicability of the connection, introducing a more functional perspective.

a link pertaining to

This emphasizes the relevance or pertinence of the link, suggesting a direct association.

an association related to

This phrase stresses the associative nature of the connection, useful in contexts where the relationship is not strictly defined.

a correlation affecting

This highlights how one element affects the other, introducing a cause-and-effect dynamic that might be absent in "a relation on".

an interrelation involving

This alternative underlines the mutual involvement of the elements in the relationship, adding a layer of complexity.

a correspondence with respect to

This option indicates the aspect or viewpoint from which the correspondence is observed, making the relationship more contextual.

an interdependence between

This highlights the mutual dependence between elements, useful in systems or networks.

a bond associated with

This alternative evokes a sense of connection or unity, altering the tone slightly.

FAQs

How is "a relation on" used in mathematical contexts?

In mathematics, "a relation on" a set A is formally defined as a subset of the Cartesian product A × A, indicating a relationship between elements within the same set.

When is it appropriate to use "a relation on" versus "a relation between"?

"A relation on" is used when the relation exists within a single set, while "a relation between" is used when the relation connects two or more distinct sets or entities. For example, you should say "a relation between sets A and B" and "a relation on set A".

What are some alternative phrases to "a relation on" that convey a similar meaning?

Depending on the context, you can use alternatives like "a relationship concerning", "a dependency with regard to", or "a connection applicable to" to express a relationship.

Can you provide an example of "a relation on" in a sentence?

Sure, here's an example: "Define a binary relation R on the set of integers such that x is related to y if x is greater than y."